\(\int \frac {\sqrt {h x} (a+b \log (c (d+e x^2)^p))}{f+g x} \, dx\) [616]

Optimal result

Integrand size = 31, antiderivative size = 1601 \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx =\text {Too large to display} \] Output:

2*a*(h*x)^(1/2)/g-8*b*p*(h*x)^(1/2)/g-2*2^(1/2)*b*d^(1/4)*h^(1/2)*p*arctan 
(1-2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/e^(1/4)/g+2*2^(1/2)*b*d^(1 
/4)*h^(1/2)*p*arctan(1+2^(1/2)*e^(1/4)*(h*x)^(1/2)/d^(1/4)/h^(1/2))/e^(1/4 
)/g+2*2^(1/2)*b*d^(1/4)*h^(1/2)*p*arctanh(2^(1/2)*d^(1/4)*e^(1/4)*(h*x)^(1 
/2)/h^(1/2)/(d^(1/2)+e^(1/2)*x))/e^(1/4)/g+2*b*(h*x)^(1/2)*ln(c*(e*x^2+d)^ 
p)/g-2*f^(1/2)*h^(1/2)*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*(a+b*ln 
(c*(e*x^2+d)^p))/g^(3/2)-8*b*f^(1/2)*h^(1/2)*p*arctan(g^(1/2)*(h*x)^(1/2)/ 
f^(1/2)/h^(1/2))*ln(2*f^(1/2)*h^(1/2)/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/ 
2)))/g^(3/2)+2*b*f^(1/2)*h^(1/2)*p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1 
/2))*ln(2*f^(1/2)*g^(1/2)*h^(1/2)*((-d)^(1/4)*(-h)^(1/2)-e^(1/4)*(h*x)^(1/ 
2))/((-d)^(1/4)*g^(1/2)*(-h)^(1/2)-I*e^(1/4)*f^(1/2)*h^(1/2))/(f^(1/2)*h^( 
1/2)-I*g^(1/2)*(h*x)^(1/2)))/g^(3/2)+2*b*f^(1/2)*h^(1/2)*p*arctan(g^(1/2)* 
(h*x)^(1/2)/f^(1/2)/h^(1/2))*ln(-2*f^(1/2)*g^(1/2)*((-d)^(1/4)*h^(1/2)-e^( 
1/4)*(h*x)^(1/2))/(I*e^(1/4)*f^(1/2)-(-d)^(1/4)*g^(1/2))/(f^(1/2)*h^(1/2)- 
I*g^(1/2)*(h*x)^(1/2)))/g^(3/2)+2*b*f^(1/2)*h^(1/2)*p*arctan(g^(1/2)*(h*x) 
^(1/2)/f^(1/2)/h^(1/2))*ln(2*f^(1/2)*g^(1/2)*h^(1/2)*((-d)^(1/4)*(-h)^(1/2 
)+e^(1/4)*(h*x)^(1/2))/((-d)^(1/4)*g^(1/2)*(-h)^(1/2)+I*e^(1/4)*f^(1/2)*h^ 
(1/2))/(f^(1/2)*h^(1/2)-I*g^(1/2)*(h*x)^(1/2)))/g^(3/2)+2*b*f^(1/2)*h^(1/2 
)*p*arctan(g^(1/2)*(h*x)^(1/2)/f^(1/2)/h^(1/2))*ln(2*f^(1/2)*g^(1/2)*((-d) 
^(1/4)*h^(1/2)+e^(1/4)*(h*x)^(1/2))/(I*e^(1/4)*f^(1/2)+(-d)^(1/4)*g^(1/...
 

Mathematica [A] (verified)

Time = 1.52 (sec) , antiderivative size = 1506, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx =\text {Too large to display} \] Input:

Integrate[(Sqrt[h*x]*(a + b*Log[c*(d + e*x^2)^p]))/(f + g*x),x]
 

Output:

(Sqrt[h*x]*(2*a*Sqrt[g]*Sqrt[x] - 8*b*Sqrt[g]*p*Sqrt[x] - (2*Sqrt[2]*b*d^( 
1/4)*Sqrt[g]*p*ArcTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)])/e^(1/4) + (2 
*Sqrt[2]*b*d^(1/4)*Sqrt[g]*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[x])/d^(1/4)] 
)/e^(1/4) - (Sqrt[2]*b*d^(1/4)*Sqrt[g]*p*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^( 
1/4)*Sqrt[x] + Sqrt[e]*x])/e^(1/4) + (Sqrt[2]*b*d^(1/4)*Sqrt[g]*p*Log[Sqrt 
[d] + Sqrt[2]*d^(1/4)*e^(1/4)*Sqrt[x] + Sqrt[e]*x])/e^(1/4) + 2*b*Sqrt[g]* 
Sqrt[x]*Log[c*(d + e*x^2)^p] + Sqrt[-f]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]]*(a 
 + b*Log[c*(d + e*x^2)^p]) - Sqrt[-f]*Log[Sqrt[-f] + Sqrt[g]*Sqrt[x]]*(a + 
 b*Log[c*(d + e*x^2)^p]) - b*Sqrt[-f]*p*(Log[(Sqrt[g]*((-d)^(1/4) - e^(1/4 
)*Sqrt[x]))/(-(e^(1/4)*Sqrt[-f]) + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqr 
t[g]*Sqrt[x]] + Log[(Sqrt[g]*((-d)^(1/4) + I*e^(1/4)*Sqrt[x]))/(I*e^(1/4)* 
Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] + Log[(Sqr 
t[g]*(I*(-d)^(1/4) + e^(1/4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^(1/4)*Sq 
rt[g])]*Log[Sqrt[-f] - Sqrt[g]*Sqrt[x]] + Log[(Sqrt[g]*((-d)^(1/4) + e^(1/ 
4)*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + (-d)^(1/4)*Sqrt[g])]*Log[Sqrt[-f] - Sqrt[ 
g]*Sqrt[x]] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*S 
qrt[-f] - (-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*S 
qrt[x]))/(e^(1/4)*Sqrt[-f] - I*(-d)^(1/4)*Sqrt[g])] + PolyLog[2, (e^(1/4)* 
(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + I*(-d)^(1/4)*Sqrt[g])] + 
 PolyLog[2, (e^(1/4)*(Sqrt[-f] - Sqrt[g]*Sqrt[x]))/(e^(1/4)*Sqrt[-f] + ...
 

Rubi [A] (verified)

Time = 4.75 (sec) , antiderivative size = 1677, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2917, 27, 2926, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx\)

\(\Big \downarrow \) 2917

\(\displaystyle \frac {2 \int \frac {h^2 x \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{f h+g x h}d\sqrt {h x}}{h}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {h x \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{f h+g x h}d\sqrt {h x}\)

\(\Big \downarrow \) 2926

\(\displaystyle 2 \int \left (\frac {a+b \log \left (c \left (e x^2+d\right )^p\right )}{g}-\frac {f h \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{g (f h+g x h)}\right )d\sqrt {h x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {\sqrt {h x} a}{g}-\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}\right )}{\sqrt [4]{e} g}+\frac {\sqrt {2} b \sqrt [4]{d} \sqrt {h} p \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} \sqrt {h x}}{\sqrt [4]{d} \sqrt {h}}+1\right )}{\sqrt [4]{e} g}+\frac {b \sqrt {h x} \log \left (c \left (e x^2+d\right )^p\right )}{g}-\frac {\sqrt {f} \sqrt {h} \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \left (a+b \log \left (c \left (e x^2+d\right )^p\right )\right )}{g^{3/2}}-\frac {4 b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{g^{3/2}}+\frac {b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}+\frac {b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}+\frac {b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}+\frac {b \sqrt {f} \sqrt {h} p \arctan \left (\frac {\sqrt {g} \sqrt {h x}}{\sqrt {f} \sqrt {h}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{g^{3/2}}-\frac {b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {e} x h+\sqrt {d} h-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right )}{\sqrt {2} \sqrt [4]{e} g}+\frac {b \sqrt [4]{d} \sqrt {h} p \log \left (\sqrt {e} x h+\sqrt {d} h+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} \sqrt {h x} \sqrt {h}\right )}{\sqrt {2} \sqrt [4]{e} g}+\frac {2 i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {h}}{\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}}\right )}{g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}-i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{2 g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}-\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}-\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}+1\right )}{2 g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \sqrt {h} \left (\sqrt [4]{-d} \sqrt {-h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (\sqrt [4]{-d} \sqrt {g} \sqrt {-h}+i \sqrt [4]{e} \sqrt {f} \sqrt {h}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{2 g^{3/2}}-\frac {i b \sqrt {f} \sqrt {h} p \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt [4]{-d} \sqrt {h}+\sqrt [4]{e} \sqrt {h x}\right )}{\left (i \sqrt [4]{e} \sqrt {f}+\sqrt [4]{-d} \sqrt {g}\right ) \left (\sqrt {f} \sqrt {h}-i \sqrt {g} \sqrt {h x}\right )}\right )}{2 g^{3/2}}-\frac {4 b p \sqrt {h x}}{g}\right )\)

Input:

Int[(Sqrt[h*x]*(a + b*Log[c*(d + e*x^2)^p]))/(f + g*x),x]
 

Output:

2*((a*Sqrt[h*x])/g - (4*b*p*Sqrt[h*x])/g - (Sqrt[2]*b*d^(1/4)*Sqrt[h]*p*Ar 
cTan[1 - (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)*Sqrt[h])])/(e^(1/4)*g) + (Sq 
rt[2]*b*d^(1/4)*Sqrt[h]*p*ArcTan[1 + (Sqrt[2]*e^(1/4)*Sqrt[h*x])/(d^(1/4)* 
Sqrt[h])])/(e^(1/4)*g) + (b*Sqrt[h*x]*Log[c*(d + e*x^2)^p])/g - (Sqrt[f]*S 
qrt[h]*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*(a + b*Log[c*(d + e*x 
^2)^p]))/g^(3/2) - (4*b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt 
[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[h])/(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h* 
x])])/g^(3/2) + (b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*S 
qrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*Sqrt[h]*((-d)^(1/4)*Sqrt[-h] - e^(1/4)*Sqr 
t[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h] - I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[ 
f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/g^(3/2) + (b*Sqrt[f]*Sqrt[h]*p*ArcTan 
[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(-2*Sqrt[f]*Sqrt[g]*((-d)^(1/4 
)*Sqrt[h] - e^(1/4)*Sqrt[h*x]))/((I*e^(1/4)*Sqrt[f] - (-d)^(1/4)*Sqrt[g])* 
(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/g^(3/2) + (b*Sqrt[f]*Sqrt[h]*p* 
ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h])]*Log[(2*Sqrt[f]*Sqrt[g]*Sqrt[ 
h]*((-d)^(1/4)*Sqrt[-h] + e^(1/4)*Sqrt[h*x]))/(((-d)^(1/4)*Sqrt[g]*Sqrt[-h 
] + I*e^(1/4)*Sqrt[f]*Sqrt[h])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h*x]))])/ 
g^(3/2) + (b*Sqrt[f]*Sqrt[h]*p*ArcTan[(Sqrt[g]*Sqrt[h*x])/(Sqrt[f]*Sqrt[h] 
)]*Log[(2*Sqrt[f]*Sqrt[g]*((-d)^(1/4)*Sqrt[h] + e^(1/4)*Sqrt[h*x]))/((I*e^ 
(1/4)*Sqrt[f] + (-d)^(1/4)*Sqrt[g])*(Sqrt[f]*Sqrt[h] - I*Sqrt[g]*Sqrt[h...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2917

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))^(p_.)]*(b_.))^(q_.)*((h_.) 
*(x_))^(m_)*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> With[{k = Denominator[ 
m]}, Simp[k/h   Subst[Int[x^(k*(m + 1) - 1)*(f + g*(x^k/h))^r*(a + b*Log[c* 
(d + e*(x^(k*n)/h^n))^p])^q, x], x, (h*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, 
 e, f, g, h, p, r}, x] && FractionQ[m] && IntegerQ[n] && IntegerQ[r]
 

rule 2926

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b 
*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c, d, e 
, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] & 
& IntegerQ[s]
 

Maple [F]

\[\int \frac {\sqrt {h x}\, \left (a +b \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )\right )}{g x +f}d x\]

Input:

int((h*x)^(1/2)*(a+b*ln(c*(e*x^2+d)^p))/(g*x+f),x)
 

Output:

int((h*x)^(1/2)*(a+b*ln(c*(e*x^2+d)^p))/(g*x+f),x)
 

Fricas [F]

\[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\int { \frac {\sqrt {h x} {\left (b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a\right )}}{g x + f} \,d x } \] Input:

integrate((h*x)^(1/2)*(a+b*log(c*(e*x^2+d)^p))/(g*x+f),x, algorithm="frica 
s")
 

Output:

integral((sqrt(h*x)*b*log((e*x^2 + d)^p*c) + sqrt(h*x)*a)/(g*x + f), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\text {Timed out} \] Input:

integrate((h*x)**(1/2)*(a+b*ln(c*(e*x**2+d)**p))/(g*x+f),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\int { \frac {\sqrt {h x} {\left (b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a\right )}}{g x + f} \,d x } \] Input:

integrate((h*x)^(1/2)*(a+b*log(c*(e*x^2+d)^p))/(g*x+f),x, algorithm="maxim 
a")
 

Output:

b*integrate((sqrt(h)*sqrt(x)*log((e*x^2 + d)^p) + sqrt(h)*sqrt(x)*log(c))/ 
(g*x + f), x) - 2*(f*h^2*arctan(sqrt(h*x)*g/sqrt(f*g*h))/(sqrt(f*g*h)*g) - 
 sqrt(h*x)*h/g)*a/h
 

Giac [F]

\[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\int { \frac {\sqrt {h x} {\left (b \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) + a\right )}}{g x + f} \,d x } \] Input:

integrate((h*x)^(1/2)*(a+b*log(c*(e*x^2+d)^p))/(g*x+f),x, algorithm="giac" 
)
 

Output:

integrate(sqrt(h*x)*(b*log((e*x^2 + d)^p*c) + a)/(g*x + f), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\int \frac {\sqrt {h\,x}\,\left (a+b\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\right )}{f+g\,x} \,d x \] Input:

int(((h*x)^(1/2)*(a + b*log(c*(d + e*x^2)^p)))/(f + g*x),x)
 

Output:

int(((h*x)^(1/2)*(a + b*log(c*(d + e*x^2)^p)))/(f + g*x), x)
 

Reduce [F]

\[ \int \frac {\sqrt {h x} \left (a+b \log \left (c \left (d+e x^2\right )^p\right )\right )}{f+g x} \, dx=\frac {\sqrt {h}\, \left (-2 e^{\frac {3}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b g p +2 e^{\frac {3}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {e}}{e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}}\right ) b g p -2 \sqrt {g}\, \sqrt {f}\, \mathit {atan} \left (\frac {\sqrt {x}\, g}{\sqrt {g}\, \sqrt {f}}\right ) a e -e^{\frac {3}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+\sqrt {d}+\sqrt {e}\, x \right ) b g p +e^{\frac {3}{4}} d^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, e^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}+\sqrt {d}+\sqrt {e}\, x \right ) b g p +2 \sqrt {x}\, a e g -4 \left (\int \frac {\sqrt {x}}{e g \,x^{4}+e f \,x^{3}+d g \,x^{2}+d f x}d x \right ) b d e f g p -4 \left (\int \frac {\sqrt {x}}{e g \,x^{3}+e f \,x^{2}+d g x +d f}d x \right ) b d e \,g^{2} p +\left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) x^{2}}{e g \,x^{3}+e f \,x^{2}+d g x +d f}d x \right ) b \,e^{2} g^{2}+\left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}{e g \,x^{3}+e f \,x^{2}+d g x +d f}d x \right ) b d e \,g^{2}\right )}{e \,g^{2}} \] Input:

int((h*x)^(1/2)*(a+b*log(c*(e*x^2+d)^p))/(g*x+f),x)
 

Output:

(sqrt(h)*( - 2*e**(3/4)*d**(1/4)*sqrt(2)*atan((e**(1/4)*d**(1/4)*sqrt(2) - 
 2*sqrt(x)*sqrt(e))/(e**(1/4)*d**(1/4)*sqrt(2)))*b*g*p + 2*e**(3/4)*d**(1/ 
4)*sqrt(2)*atan((e**(1/4)*d**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(e))/(e**(1/4)* 
d**(1/4)*sqrt(2)))*b*g*p - 2*sqrt(g)*sqrt(f)*atan((sqrt(x)*g)/(sqrt(g)*sqr 
t(f)))*a*e - e**(3/4)*d**(1/4)*sqrt(2)*log( - sqrt(x)*e**(1/4)*d**(1/4)*sq 
rt(2) + sqrt(d) + sqrt(e)*x)*b*g*p + e**(3/4)*d**(1/4)*sqrt(2)*log(sqrt(x) 
*e**(1/4)*d**(1/4)*sqrt(2) + sqrt(d) + sqrt(e)*x)*b*g*p + 2*sqrt(x)*a*e*g 
- 4*int(sqrt(x)/(d*f*x + d*g*x**2 + e*f*x**3 + e*g*x**4),x)*b*d*e*f*g*p - 
4*int(sqrt(x)/(d*f + d*g*x + e*f*x**2 + e*g*x**3),x)*b*d*e*g**2*p + int((s 
qrt(x)*log((d + e*x**2)**p*c)*x**2)/(d*f + d*g*x + e*f*x**2 + e*g*x**3),x) 
*b*e**2*g**2 + int((sqrt(x)*log((d + e*x**2)**p*c))/(d*f + d*g*x + e*f*x** 
2 + e*g*x**3),x)*b*d*e*g**2))/(e*g**2)